| // Copyright ©2017 The gonum Authors. All rights reserved. |
| // Use of this source code is governed by a BSD-style |
| // license that can be found in the LICENSE file. |
| |
| package testlapack |
| |
| import ( |
| "math" |
| "math/rand" |
| |
| "gonum.org/v1/gonum/blas" |
| "gonum.org/v1/gonum/blas/blas64" |
| ) |
| |
| // Dlatm1 computes the entries of dst as specified by mode, cond and rsign. |
| // |
| // mode describes how dst will be computed: |
| // |mode| == 1: dst[0] = 1 and dst[1:n] = 1/cond |
| // |mode| == 2: dst[:n-1] = 1/cond and dst[n-1] = 1 |
| // |mode| == 3: dst[i] = cond^{-i/(n-1)}, i=0,...,n-1 |
| // |mode| == 4: dst[i] = 1 - i*(1-1/cond)/(n-1) |
| // |mode| == 5: dst[i] = random number in the range (1/cond, 1) such that |
| // their logarithms are uniformly distributed |
| // |mode| == 6: dst[i] = random number from the distribution given by dist |
| // If mode is negative, the order of the elements of dst will be reversed. |
| // For other values of mode Dlatm1 will panic. |
| // |
| // If rsign is true and mode is not ±6, each entry of dst will be multiplied by 1 |
| // or -1 with probability 0.5 |
| // |
| // dist specifies the type of distribution to be used when mode == ±6: |
| // dist == 1: Uniform[0,1) |
| // dist == 2: Uniform[-1,1) |
| // dist == 3: Normal(0,1) |
| // For other values of dist Dlatm1 will panic. |
| // |
| // rnd is used as a source of random numbers. |
| func Dlatm1(dst []float64, mode int, cond float64, rsign bool, dist int, rnd *rand.Rand) { |
| amode := mode |
| if amode < 0 { |
| amode = -amode |
| } |
| if amode < 1 || 6 < amode { |
| panic("testlapack: invalid mode") |
| } |
| if cond < 1 { |
| panic("testlapack: cond < 1") |
| } |
| if amode == 6 && (dist < 1 || 3 < dist) { |
| panic("testlapack: invalid dist") |
| } |
| |
| n := len(dst) |
| if n == 0 { |
| return |
| } |
| |
| switch amode { |
| case 1: |
| dst[0] = 1 |
| for i := 1; i < n; i++ { |
| dst[i] = 1 / cond |
| } |
| case 2: |
| for i := 0; i < n-1; i++ { |
| dst[i] = 1 |
| } |
| dst[n-1] = 1 / cond |
| case 3: |
| dst[0] = 1 |
| if n > 1 { |
| alpha := math.Pow(cond, -1/float64(n-1)) |
| for i := 1; i < n; i++ { |
| dst[i] = math.Pow(alpha, float64(i)) |
| } |
| } |
| case 4: |
| dst[0] = 1 |
| if n > 1 { |
| condInv := 1 / cond |
| alpha := (1 - condInv) / float64(n-1) |
| for i := 1; i < n; i++ { |
| dst[i] = float64(n-i-1)*alpha + condInv |
| } |
| } |
| case 5: |
| alpha := math.Log(1 / cond) |
| for i := range dst { |
| dst[i] = math.Exp(alpha * rnd.Float64()) |
| } |
| case 6: |
| switch dist { |
| case 1: |
| for i := range dst { |
| dst[i] = rnd.Float64() |
| } |
| case 2: |
| for i := range dst { |
| dst[i] = 2*rnd.Float64() - 1 |
| } |
| case 3: |
| for i := range dst { |
| dst[i] = rnd.NormFloat64() |
| } |
| } |
| } |
| |
| if rsign && amode != 6 { |
| for i, v := range dst { |
| if rnd.Float64() < 0.5 { |
| dst[i] = -v |
| } |
| } |
| } |
| |
| if mode < 0 { |
| for i := 0; i < n/2; i++ { |
| dst[i], dst[n-i-1] = dst[n-i-1], dst[i] |
| } |
| } |
| } |
| |
| // Dlagsy generates an n×n symmetric matrix A, by pre- and post- multiplying a |
| // real diagonal matrix D with a random orthogonal matrix: |
| // A = U * D * U^T. |
| // |
| // work must have length at least 2*n, otherwise Dlagsy will panic. |
| // |
| // The parameter k is unused but it must satisfy |
| // 0 <= k <= n-1. |
| func Dlagsy(n, k int, d []float64, a []float64, lda int, rnd *rand.Rand, work []float64) { |
| checkMatrix(n, n, a, lda) |
| if k < 0 || max(0, n-1) < k { |
| panic("testlapack: invalid value of k") |
| } |
| if len(d) != n { |
| panic("testlapack: bad length of d") |
| } |
| if len(work) < 2*n { |
| panic("testlapack: insufficient work length") |
| } |
| |
| // Initialize lower triangle of A to diagonal matrix. |
| for i := 1; i < n; i++ { |
| for j := 0; j < i; j++ { |
| a[i*lda+j] = 0 |
| } |
| } |
| for i := 0; i < n; i++ { |
| a[i*lda+i] = d[i] |
| } |
| |
| bi := blas64.Implementation() |
| |
| // Generate lower triangle of symmetric matrix. |
| for i := n - 2; i >= 0; i-- { |
| for j := 0; j < n-i; j++ { |
| work[j] = rnd.NormFloat64() |
| } |
| wn := bi.Dnrm2(n-i, work[:n-i], 1) |
| wa := math.Copysign(wn, work[0]) |
| var tau float64 |
| if wn != 0 { |
| wb := work[0] + wa |
| bi.Dscal(n-i-1, 1/wb, work[1:n-i], 1) |
| work[0] = 1 |
| tau = wb / wa |
| } |
| |
| // Apply random reflection to A[i:n,i:n] from the left and the |
| // right. |
| // |
| // Compute y := tau * A * u. |
| bi.Dsymv(blas.Lower, n-i, tau, a[i*lda+i:], lda, work[:n-i], 1, 0, work[n:2*n-i], 1) |
| |
| // Compute v := y - 1/2 * tau * ( y, u ) * u. |
| alpha := -0.5 * tau * bi.Ddot(n-i, work[n:2*n-i], 1, work[:n-i], 1) |
| bi.Daxpy(n-i, alpha, work[:n-i], 1, work[n:2*n-i], 1) |
| |
| // Apply the transformation as a rank-2 update to A[i:n,i:n]. |
| bi.Dsyr2(blas.Lower, n-i, -1, work[:n-i], 1, work[n:2*n-i], 1, a[i*lda+i:], lda) |
| } |
| |
| // Store full symmetric matrix. |
| for i := 1; i < n; i++ { |
| for j := 0; j < i; j++ { |
| a[j*lda+i] = a[i*lda+j] |
| } |
| } |
| } |
| |
| // Dlagge generates a real general m×n matrix A, by pre- and post-multiplying |
| // a real diagonal matrix D with random orthogonal matrices: |
| // A = U*D*V. |
| // |
| // d must have length min(m,n), and work must have length m+n, otherwise Dlagge |
| // will panic. |
| // |
| // The parameters ku and kl are unused but they must satisfy |
| // 0 <= kl <= m-1, |
| // 0 <= ku <= n-1. |
| func Dlagge(m, n, kl, ku int, d []float64, a []float64, lda int, rnd *rand.Rand, work []float64) { |
| checkMatrix(m, n, a, lda) |
| if kl < 0 || max(0, m-1) < kl { |
| panic("testlapack: invalid value of kl") |
| } |
| if ku < 0 || max(0, n-1) < ku { |
| panic("testlapack: invalid value of ku") |
| } |
| if len(d) != min(m, n) { |
| panic("testlapack: bad length of d") |
| } |
| if len(work) < m+n { |
| panic("testlapack: insufficient work length") |
| } |
| |
| // Initialize A to diagonal matrix. |
| for i := 0; i < m; i++ { |
| for j := 0; j < n; j++ { |
| a[i*lda+j] = 0 |
| } |
| } |
| for i := 0; i < min(m, n); i++ { |
| a[i*lda+i] = d[i] |
| } |
| |
| // Quick exit if the user wants a diagonal matrix. |
| // if kl == 0 && ku == 0 { |
| // return |
| // } |
| |
| bi := blas64.Implementation() |
| |
| // Pre- and post-multiply A by random orthogonal matrices. |
| for i := min(m, n) - 1; i >= 0; i-- { |
| if i < m-1 { |
| for j := 0; j < m-i; j++ { |
| work[j] = rnd.NormFloat64() |
| } |
| wn := bi.Dnrm2(m-i, work[:m-i], 1) |
| wa := math.Copysign(wn, work[0]) |
| var tau float64 |
| if wn != 0 { |
| wb := work[0] + wa |
| bi.Dscal(m-i-1, 1/wb, work[1:m-i], 1) |
| work[0] = 1 |
| tau = wb / wa |
| } |
| |
| // Multiply A[i:m,i:n] by random reflection from the left. |
| bi.Dgemv(blas.Trans, m-i, n-i, |
| 1, a[i*lda+i:], lda, work[:m-i], 1, |
| 0, work[m:m+n-i], 1) |
| bi.Dger(m-i, n-i, |
| -tau, work[:m-i], 1, work[m:m+n-i], 1, |
| a[i*lda+i:], lda) |
| } |
| if i < n-1 { |
| for j := 0; j < n-i; j++ { |
| work[j] = rnd.NormFloat64() |
| } |
| wn := bi.Dnrm2(n-i, work[:n-i], 1) |
| wa := math.Copysign(wn, work[0]) |
| var tau float64 |
| if wn != 0 { |
| wb := work[0] + wa |
| bi.Dscal(n-i-1, 1/wb, work[1:n-i], 1) |
| work[0] = 1 |
| tau = wb / wa |
| } |
| |
| // Multiply A[i:m,i:n] by random reflection from the right. |
| bi.Dgemv(blas.NoTrans, m-i, n-i, |
| 1, a[i*lda+i:], lda, work[:n-i], 1, |
| 0, work[n:n+m-i], 1) |
| bi.Dger(m-i, n-i, |
| -tau, work[n:n+m-i], 1, work[:n-i], 1, |
| a[i*lda+i:], lda) |
| } |
| } |
| |
| // TODO(vladimir-ch): Reduce number of subdiagonals to kl and number of |
| // superdiagonals to ku. |
| } |
| |
| // dlarnv fills dst with random numbers from a uniform or normal distribution |
| // specified by dist: |
| // dist=1: uniform(0,1), |
| // dist=2: uniform(-1,1), |
| // dist=3: normal(0,1). |
| // For other values of dist dlarnv will panic. |
| func dlarnv(dst []float64, dist int, rnd *rand.Rand) { |
| switch dist { |
| default: |
| panic("testlapack: invalid dist") |
| case 1: |
| for i := range dst { |
| dst[i] = rnd.Float64() |
| } |
| case 2: |
| for i := range dst { |
| dst[i] = 2*rnd.Float64() - 1 |
| } |
| case 3: |
| for i := range dst { |
| dst[i] = rnd.NormFloat64() |
| } |
| } |
| } |
| |
| // dlattr generates an n×n triangular test matrix A with its properties uniquely |
| // determined by imat and uplo, and returns whether A has unit diagonal. If diag |
| // is blas.Unit, the diagonal elements are set so that A[k,k]=k. |
| // |
| // trans specifies whether the matrix A or its transpose will be used. |
| // |
| // If imat is greater than 10, dlattr also generates the right hand side of the |
| // linear system A*x=b, or A^T*x=b. Valid values of imat are 7, and all between 11 |
| // and 19, inclusive. |
| // |
| // b mush have length n, and work must have length 3*n, and dlattr will panic |
| // otherwise. |
| func dlattr(imat int, uplo blas.Uplo, trans blas.Transpose, n int, a []float64, lda int, b, work []float64, rnd *rand.Rand) (diag blas.Diag) { |
| checkMatrix(n, n, a, lda) |
| if len(b) != n { |
| panic("testlapack: bad length of b") |
| } |
| if len(work) < 3*n { |
| panic("testlapack: insufficient length of work") |
| } |
| if uplo != blas.Upper && uplo != blas.Lower { |
| panic("testlapack: bad uplo") |
| } |
| if trans != blas.Trans && trans != blas.NoTrans { |
| panic("testlapack: bad trans") |
| } |
| |
| if n == 0 { |
| return blas.NonUnit |
| } |
| |
| ulp := dlamchE * dlamchB |
| smlnum := dlamchS |
| bignum := (1 - ulp) / smlnum |
| |
| bi := blas64.Implementation() |
| |
| switch imat { |
| default: |
| // TODO(vladimir-ch): Implement the remaining cases. |
| panic("testlapack: invalid or unimplemented imat") |
| case 7: |
| // Identity matrix. The diagonal is set to NaN. |
| diag = blas.Unit |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| a[i*lda+i] = math.NaN() |
| for j := i + 1; j < n; j++ { |
| a[i*lda+j] = 0 |
| } |
| } |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| for j := 0; j < i; j++ { |
| a[i*lda+j] = 0 |
| } |
| a[i*lda+i] = math.NaN() |
| } |
| } |
| case 11: |
| // Generate a triangular matrix with elements between -1 and 1, |
| // give the diagonal norm 2 to make it well-conditioned, and |
| // make the right hand side large so that it requires scaling. |
| diag = blas.NonUnit |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n-1; i++ { |
| dlarnv(a[i*lda+i:i*lda+n], 2, rnd) |
| } |
| case blas.Lower: |
| for i := 1; i < n; i++ { |
| dlarnv(a[i*lda:i*lda+i+1], 2, rnd) |
| } |
| } |
| for i := 0; i < n; i++ { |
| a[i*lda+i] = math.Copysign(2, a[i*lda+i]) |
| } |
| // Set the right hand side so that the largest value is bignum. |
| dlarnv(b, 2, rnd) |
| imax := bi.Idamax(n, b, 1) |
| bscal := bignum / math.Max(1, b[imax]) |
| bi.Dscal(n, bscal, b, 1) |
| case 12: |
| // Make the first diagonal element in the solve small to cause |
| // immediate overflow when dividing by T[j,j]. The off-diagonal |
| // elements are small (cnorm[j] < 1). |
| diag = blas.NonUnit |
| tscal := 1 / math.Max(1, float64(n-1)) |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda+i:i*lda+n], 2, rnd) |
| bi.Dscal(n-i-1, tscal, a[i*lda+i+1:], 1) |
| a[i*lda+i] = math.Copysign(1, a[i*lda+i]) |
| } |
| a[(n-1)*lda+n-1] *= smlnum |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda:i*lda+i+1], 2, rnd) |
| bi.Dscal(i, tscal, a[i*lda:], 1) |
| a[i*lda+i] = math.Copysign(1, a[i*lda+i]) |
| } |
| a[0] *= smlnum |
| } |
| dlarnv(b, 2, rnd) |
| case 13: |
| // Make the first diagonal element in the solve small to cause |
| // immediate overflow when dividing by T[j,j]. The off-diagonal |
| // elements are O(1) (cnorm[j] > 1). |
| diag = blas.NonUnit |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda+i:i*lda+n], 2, rnd) |
| a[i*lda+i] = math.Copysign(1, a[i*lda+i]) |
| } |
| a[(n-1)*lda+n-1] *= smlnum |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda:i*lda+i+1], 2, rnd) |
| a[i*lda+i] = math.Copysign(1, a[i*lda+i]) |
| } |
| a[0] *= smlnum |
| } |
| dlarnv(b, 2, rnd) |
| case 14: |
| // T is diagonal with small numbers on the diagonal to |
| // make the growth factor underflow, but a small right hand side |
| // chosen so that the solution does not overflow. |
| diag = blas.NonUnit |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| for j := i + 1; j < n; j++ { |
| a[i*lda+j] = 0 |
| } |
| if (n-1-i)&0x2 == 0 { |
| a[i*lda+i] = smlnum |
| } else { |
| a[i*lda+i] = 1 |
| } |
| } |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| for j := 0; j < i; j++ { |
| a[i*lda+j] = 0 |
| } |
| if i&0x2 == 0 { |
| a[i*lda+i] = smlnum |
| } else { |
| a[i*lda+i] = 1 |
| } |
| } |
| } |
| // Set the right hand side alternately zero and small. |
| switch uplo { |
| case blas.Upper: |
| b[0] = 0 |
| for i := n - 1; i > 0; i -= 2 { |
| b[i] = 0 |
| b[i-1] = smlnum |
| } |
| case blas.Lower: |
| for i := 0; i < n-1; i += 2 { |
| b[i] = 0 |
| b[i+1] = smlnum |
| } |
| b[n-1] = 0 |
| } |
| case 15: |
| // Make the diagonal elements small to cause gradual overflow |
| // when dividing by T[j,j]. To control the amount of scaling |
| // needed, the matrix is bidiagonal. |
| diag = blas.NonUnit |
| texp := 1 / math.Max(1, float64(n-1)) |
| tscal := math.Pow(smlnum, texp) |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| a[i*lda+i] = tscal |
| if i < n-1 { |
| a[i*lda+i+1] = -1 |
| } |
| for j := i + 2; j < n; j++ { |
| a[i*lda+j] = 0 |
| } |
| } |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| for j := 0; j < i-1; j++ { |
| a[i*lda+j] = 0 |
| } |
| if i > 0 { |
| a[i*lda+i-1] = -1 |
| } |
| a[i*lda+i] = tscal |
| } |
| } |
| dlarnv(b, 2, rnd) |
| case 16: |
| // One zero diagonal element. |
| diag = blas.NonUnit |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda+i:i*lda+n], 2, rnd) |
| a[i*lda+i] = math.Copysign(2, a[i*lda+i]) |
| } |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda:i*lda+i+1], 2, rnd) |
| a[i*lda+i] = math.Copysign(2, a[i*lda+i]) |
| } |
| } |
| iy := n / 2 |
| a[iy*lda+iy] = 0 |
| dlarnv(b, 2, rnd) |
| bi.Dscal(n, 2, b, 1) |
| case 17: |
| // Make the offdiagonal elements large to cause overflow when |
| // adding a column of T. In the non-transposed case, the matrix |
| // is constructed to cause overflow when adding a column in |
| // every other step. |
| diag = blas.NonUnit |
| tscal := (1 - ulp) / (dlamchS / ulp) |
| texp := 1.0 |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| for j := i; j < n; j++ { |
| a[i*lda+j] = 0 |
| } |
| } |
| for j := n - 1; j >= 1; j -= 2 { |
| a[j] = -tscal / float64(n+1) |
| a[j*lda+j] = 1 |
| b[j] = texp * (1 - ulp) |
| a[j-1] = -tscal / float64(n+1) / float64(n+2) |
| a[(j-1)*lda+j-1] = 1 |
| b[j-1] = texp * float64(n*n+n-1) |
| texp *= 2 |
| } |
| b[0] = float64(n+1) / float64(n+2) * tscal |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| for j := 0; j <= i; j++ { |
| a[i*lda+j] = 0 |
| } |
| } |
| for j := 0; j < n-1; j += 2 { |
| a[(n-1)*lda+j] = -tscal / float64(n+1) |
| a[j*lda+j] = 1 |
| b[j] = texp * (1 - ulp) |
| a[(n-1)*lda+j+1] = -tscal / float64(n+1) / float64(n+2) |
| a[(j+1)*lda+j+1] = 1 |
| b[j+1] = texp * float64(n*n+n-1) |
| texp *= 2 |
| } |
| b[n-1] = float64(n+1) / float64(n+2) * tscal |
| } |
| case 18: |
| // Generate a unit triangular matrix with elements between -1 |
| // and 1, and make the right hand side large so that it requires |
| // scaling. The diagonal is set to NaN. |
| diag = blas.Unit |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| a[i*lda+i] = math.NaN() |
| dlarnv(a[i*lda+i+1:i*lda+n], 2, rnd) |
| } |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda:i*lda+i], 2, rnd) |
| a[i*lda+i] = math.NaN() |
| } |
| } |
| // Set the right hand side so that the largest value is bignum. |
| dlarnv(b, 2, rnd) |
| iy := bi.Idamax(n, b, 1) |
| bnorm := math.Abs(b[iy]) |
| bscal := bignum / math.Max(1, bnorm) |
| bi.Dscal(n, bscal, b, 1) |
| case 19: |
| // Generate a triangular matrix with elements between |
| // bignum/(n-1) and bignum so that at least one of the column |
| // norms will exceed bignum. |
| // Dlatrs cannot handle this case for (typically) n>5. |
| diag = blas.NonUnit |
| tleft := bignum / math.Max(1, float64(n-1)) |
| tscal := bignum * (float64(n-1) / math.Max(1, float64(n))) |
| switch uplo { |
| case blas.Upper: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda+i:i*lda+n], 2, rnd) |
| for j := i; j < n; j++ { |
| aij := a[i*lda+j] |
| a[i*lda+j] = math.Copysign(tleft, aij) + tscal*aij |
| } |
| } |
| case blas.Lower: |
| for i := 0; i < n; i++ { |
| dlarnv(a[i*lda:i*lda+i+1], 2, rnd) |
| for j := 0; j <= i; j++ { |
| aij := a[i*lda+j] |
| a[i*lda+j] = math.Copysign(tleft, aij) + tscal*aij |
| } |
| } |
| } |
| dlarnv(b, 2, rnd) |
| bi.Dscal(n, 2, b, 1) |
| } |
| |
| // Flip the matrix if the transpose will be used. |
| if trans == blas.Trans { |
| switch uplo { |
| case blas.Upper: |
| for j := 0; j < n/2; j++ { |
| bi.Dswap(n-2*j-1, a[j*lda+j:], 1, a[(j+1)*lda+n-j-1:], -lda) |
| } |
| case blas.Lower: |
| for j := 0; j < n/2; j++ { |
| bi.Dswap(n-2*j-1, a[j*lda+j:], lda, a[(n-j-1)*lda+j+1:], -1) |
| } |
| } |
| } |
| |
| return diag |
| } |
| |
| func checkMatrix(m, n int, a []float64, lda int) { |
| if m < 0 { |
| panic("testlapack: m < 0") |
| } |
| if n < 0 { |
| panic("testlapack: n < 0") |
| } |
| if lda < max(1, n) { |
| panic("testlapack: lda < max(1, n)") |
| } |
| if len(a) < (m-1)*lda+n { |
| panic("testlapack: insufficient matrix slice length") |
| } |
| } |